Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

Abstract

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg-Tran-Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane.

Figures (7)

• Figure 1: Lifting an $(n,m)$-shuffle to a braid.
• Figure 2: A configuration in Conf$_{(2,3,5,3)} (\mathbb{C}) \subset$ Conf$_{13} (\mathbb{C})$. The configuration is mapped by $\pi$ to $(x_1, x_1, x_2, x_2, x_2, x_3, x_3, x_3, x_3, x_3, x_4, x_4, x_4) \in \mathrm{Sym}_{(2,3,5,3)} (\mathbb{R}) \subset \mathrm{Sym}_{13} (\mathbb{R})$. Geometrically, the points in the configuration are arranged into columns based on their (ordered) real coordinates, while the configuration factors $\mathrm{Conf}_{\lambda_i}(\mathbb{R})$ record the imaginary parts of the points in respective columns.
• Figure 3: Decomposition of an $((m,h),n,n-1)$-shuffle $\gamma$ into a sequence of two permutations: the permutation $\omega$, followed by an $(h,n-1)$-shuffle $\beta$ on the integer interval $\llbracket 1, n+h-1 \rrbracket$. The lift $\tilde{\gamma}$ is the product of lifts of the component permutations to $B_{n+m}$, i.e. $\tilde{\gamma} = \tilde{\beta} \tilde{\omega}$. The factor $w$ (in bold) is sent from $n+m$ to $n+h$.
• Figure 4: Decomposition of an $(n,(m,h),n-1)$-shuffle $\gamma$ into a sequence of two permutations: the permutation $\omega$, followed by a $(n-1,h)$-shuffle $\beta$ on the integer interval $\llbracket 1,n+h-1 \rrbracket$. The lift $\tilde{\gamma}$ is given by $\tilde{\gamma} = \tilde{\beta}\tilde{\omega}$. The factor $w$ (in bold) is mapped from $n$ to $n+h$.
• Figure 5: A configuration in $e_{((2,3,4,5,3),3,1)} \subset$ Conf$_{16} (\mathbb{C}^\times)$. The removed origin $O$ (mapped to a fixed point in the embedded image of Conf$_{16} (\mathbb{C}^\times)$ in Conf$_{17} (\mathbb{C})$) lies on the third vertical column from the left with one point below.

Theorems & Definitions (76)

• Theorem 1.1: etw17
• Theorem 1.2
• Definition 2.1
• Proposition 2.2: etw17
• Definition 2.3
• Proposition 2.4: cri99
• Definition 2.5
• Proposition 2.6
• Example 2.7
• Example 2.8